Determining the Hurwitz orbit of the standard generators of a braid group

Abstract

The Hurwitz action of the $n$-braid group $B_{n}$ on the $n$-fold product $(B_{m})^{n}$ of the $m$-braid group $B_{m}$ is studied. Using a natural action of $B_{n}$ on trees with $n$ labeled edges and $n+1$ labeled vertices, we determine all elements of the orbit of every $n$-tuple of the $n$ distinct standard generators of $B_{n+1}$ under the Hurwitz action of $B_{n}$.